Irrationalité de valeurs de zêta (d'après Apéry, Rivoal, ...)

Details Irrationalité de valeurs de zêta (d'après Apéry, Rivoal, ...) Irrationalité de valeurs de zêta (d'après Apéry, Rivoal, ...)

2003-03-05

arxiv.org/abs/math/0303066v1

S\'eminaire Bourbaki 2002-2003 expos\'e no. 910 (Nov. 2002); Ast\'erisque 294 (2004), 27-62

Mathematics

Number Theory

Bourbaki Seminar, November 2002 ; to appear in Ast\'erisque ; 45 pages ; in french

Scientific paper

This survey text deals with irrationality, and linear independence over the rationals, of values at positive odd integers of Riemann zeta function. The first section gives all known proofs (and connections between them) of Ap\'ery's Theorem (1978) : $\zeta(3)$ is irrational. The second section is devoted to a variant of the proof, published by Rivoal and Ball-Rivoal, that infinitely many $\zeta(2n+1)$ are irrational. The end of this text deals with more quantitative statements.

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